Simplify the following expression and state the condition under which the simplification is valid: $t = \dfrac{r^2 - r - 42}{r^2 + 6r}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{r^2 - r - 42}{r^2 + 6r} = \dfrac{(r - 7)(r + 6)}{(r)(r + 6)} $ Notice that the term $(r + 6)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(r + 6)$ gives: $t = \dfrac{r - 7}{r}$ Since we divided by $(r + 6)$, $r \neq -6$. $t = \dfrac{r - 7}{r}; \space r \neq -6$